Positivity and Transportation

TL;DR

This paper introduces a new positive definite kernel for transportation matrices between histograms, focusing on a subset of solutions to enable efficient computation while maintaining mathematical properties.

Contribution

It proposes an alternative kernel based on Northwestern corner solutions, reducing computational complexity and ensuring positive definiteness.

Findings

Kernel is positive definite and computationally efficient.

Sampling a subset of vertices controls the kernel's complexity.

The method is applicable to histograms of modest dimensions.

Abstract

We prove in this paper that the weighted volume of the set of integral transportation matrices between two integral histograms r and c of equal sum is a positive definite kernel of r and c when the set of considered weights forms a positive definite matrix. The computation of this quantity, despite being the subject of a significant research effort in algebraic statistics, remains an intractable challenge for histograms of even modest dimensions. We propose an alternative kernel which, rather than considering all matrices of the transportation polytope, only focuses on a sub-sample of its vertices known as its Northwestern corner solutions. The resulting kernel is positive definite and can be computed with a number of operations O(R^2d) that grows linearly in the complexity of the dimension d, where R^2, the total amount of sampled vertices, is a parameter that controls the complexity of the kernel.